It's more in the hands of your school sending the transcripts than when you submit the application. Of course you don't want to have your transcripts sitting at AADSAS waiting on transcripts. Your school might give you a timeframe on processing time. If they stick to it and you get them the forms quickly, you should be okay (early) as long as you submit before or close to the time they arrive at AADSAS.
They say it takes several weeks to process and verify, but it'll probably be faster early on (not guaranteed of course). My transcripts were verified in 3 days once they both arrived. That made first batch easily. I think first batch has been 4-5 weeks out from the cycle opening in the past. If you don't make first batch, you'll be fine too.